Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [predicate-logic]

Questions concerning predicate calculus, i.e. the logic of quantifiers.

Some well-known formal systems covered by this term are

  • first-order logic, containing the quantifiers $\forall$ and $\exists$
  • second-order logic
  • many-sorted logic
  • infinitary logic
4144 questions
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Factoring quantifiers.. How is P(x)∧ ∀xQ(x) logically equivalent to ∀x(P(x)∧Q(x))?

Trying to learn about predicate formulas. I was told that (let E and F be predicates and Q be a quantifier ∃ or ∀) E ∧ QxF is logically equivalent to Qx(E∧F), if x is not free in E So let's take the formula below: P(x) ∧ ∀xQ(x) is logically…
user2719875
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Predicate Logic / changing the result of a statement by asking statment?

Let's have a simple statement that P:(x) is false It's good weather today. (it's not) But by evoking the statement, the result can change. Is there something in any field of Mathematics that describe this case? Or it's completely out of rules? I'm…
Martin Brisiak
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What determines a predicate wff's correctness?

Write the English language statement as a predicate wff (the domain of interpretation is the whole world). All games are more fun than some movie. G(x): x is a game M(x): x is a movie F(x,y): x is more fun than y My attempt: $$(\forall x)(\exists…
Parabolord
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Predicate Logic Well Formed Formula: Is ((∀x P(x)) ∨ (∃y Q(f (y)))) well formed?

Having difficulties understanding well formed formulas. I understand rules, but don't know how they apply to these. If they're not well formed formulas, how do I write it so that it'll be a well formed predicate formula. Please help. 1) ((∀x P(x)) ∨…
JiHua
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tree method question

When using the tree method for testing the consistency of a set of sentences, if I have the set of equivalent sentences {(∃x)¬Fx , ¬(∀x)Fx)}, can one fully develop a tree with only the constant a, or do I have to extend it using three constants a, b…
Thrasymaque
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predicate logic - function mapping outside of domain

Can a model of a sentence in predicate logic contain a function that can map to a member not contained in the domain of that model? Example. Is this interpretation correct : domain= $\{1,2\}$ ; sentence=$'(∃x)(∃y)(P(f(xy)))'$ ; $f(xy) =$ the the…
Thrasymaque
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Composition computation

I wondered if the computational steps works for all different kind of languages in different word orders? such as SVO, VOS, VSO. Thank you!!!
Lisa
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valid consequence model

I just can't wrap my head around why this statement does not hold: ∃x∃yRxy ⊨ ∃x∃yRyx Let D = {1, 2} and Let I(R) = {(1,2)} M ⊨ ∃x∃yRxy = 1, since some x(1) has some y(2) such that (x,y) ∈ I(R) But in my eyes: M ⊨ ∃x∃yRyx = 1, since some x(2) has…
Char
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Prove by tableaux that theory T' is unsatifiable

I have to find a new theory by skolemization and prove that it is incosistent by tableau method. My result: My question is, if I have one true and one false sign with same relation and the elements are not in the same place, does it lead to…
Muffy
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Is this formalization correct? If not, why?

Is this formalization correct, and if it's not, why? And what would be the correct formalization? I am assuming Russell's theory of definite descriptions. The author of The Da Vinci Code is the author of Angels and Demons $$\exists x(P(x) \wedge…
user405159
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(Predicate Logic) ∀x ∈ N(x > 1 → ∃k ∈ N∃m ∈ N(m ≡ 1(mod 2) ∧ x = 2^k⋅m))

Can anyone show me how to prove this statement? I'm very confused. I have tried substitute numbers into each x, k and m, it seems this statement is true but it needs to be proved by mathematical methods. Thank you!
jhg
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Show that $k= \{\mathfrak A|\mathfrak A\text{ is S-model and} \space R^\mathfrak{A} \space \text{is well founded}\}$ is not $\Delta$-elementary class

I need to show that the set: $k= \{\mathfrak A|\mathfrak A\text{ is S-model and} \space R^\mathfrak{a} \space \text{is well founded}\}$ is not a $\Delta$-elementary class, that is: $k \neq \{\mathfrak A|\mathfrak A\text{ is S-model and} \space…
Dole
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How to build the syntactic tree of this formula

Consider the following formula: ∃x(Ux & ∀y((Ay & Ixy) ⊃ Fxy)) I would like to build the syntactic tree of this formula but I'm getting confused about how to do it. In particular, I don't know how to treat operators like ∃: do I have to treat them…
user405159
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Predicate logic: free variable after ∀

Consider the formula ∀xα (where x is some variable). Is the occurrence of x which comes just after ∀ considered as a free variable? Many thanks for your help. Fish
user405159
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Do completeness and correctness theorems apply to inconsistent sets in predicate logic?

Simple question, the way I understand the theorems, the result is that the completeness and correctness theorems apply to any set. However, consider the set of formulas: $\phi, \neg \phi$ Or any other inconsistent set. Then it is commonly known…
Dole
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